Porisma

Worked solution

Compare curves with their reciprocals

Use a graphing calculator to sketch (a)(i) y=x+2y=x+2, (ii) y=1/(x+2)y=1/(x+2); (b)(i) y=(x+1)(x−3)y=(x+1)(x-3), (ii) y=1/((x+1)(x−3))y=1/((x+1)(x-3)). By comparing each pair, what do you observe about the relationship between y=f(x)y=f(x) and y=1/f(x)y=1/f(x)?

CJC · Transformations · Reciprocal activity

(a) Sketch the line and its reciprocal

Answer

The reciprocal has asymptotes x=−2x=-2, y=0y=0 and y-intercept (0,1/2)(0,1/2).

The original line has intercepts (−2,0)(-2,0) and (0,2)(0,2). The Teacher fills the blank Student reciprocal axes. Invert each non-zero output, preserving its sign; exclude x=−2x=-2. The reciprocal has no x-intercept.

Expand diagram

Use the scrollbars or arrow keys to inspect details.

Separate line x+2 through (−2,0),(0,2), and reciprocal 1/(x+2), asymptotes x=−2,y=0, y-intercept (0,1/2), with two branches and no x-intercept.
Separate line x+2 through (−2,0),(0,2), and reciprocal 1/(x+2), asymptotes x=−2,y=0, y-intercept (0,1/2), with two branches and no x-intercept.
Separate line x+2 through (−2,0),(0,2), and reciprocal 1/(x+2), asymptotes x=−2,y=0, y-intercept (0,1/2), with two branches and no x-intercept.
How did this part go?

(b) Sketch the quadratic and its reciprocal

Answer

The reciprocal has asymptotes x=−1x=-1, x=3x=3, y=0y=0; y-intercept (0,−1/3)(0,-1/3) and local maximum (1,−1/4)(1,-1/4).

The original roots are −1,3-1,3, y-intercept is −3-3 and minimum is (1,−4)(1,-4). The Teacher completes the reciprocal graph with all these transformed labels. Preserve three branches: positive outside the roots, negative between them. The negative original minimum becomes a negative reciprocal maximum.

Expand diagram

Use the scrollbars or arrow keys to inspect details.

Separate quadratic with roots −1,3, minimum (1,−4), y-intercept −3, and reciprocal with poles −1,3, horizontal asymptote 0, y-intercept −1/3 and maximum (1,−1/4). Three reciprocal branches preserve the signs.
Separate quadratic with roots −1,3, minimum (1,−4), y-intercept −3, and reciprocal with poles −1,3, horizontal asymptote 0, y-intercept −1/3 and maximum (1,−1/4). Three reciprocal branches preserve the signs.
Separate quadratic with roots −1,3, minimum (1,−4), y-intercept −3, and reciprocal with poles −1,3, horizontal asymptote 0, y-intercept −1/3 and maximum (1,−1/4). Three reciprocal branches preserve the signs.
How did this part go?

Compare signs, extrema and excluded inputs

Answer

(x,y)↦(x,1/y)(x,y)\mapsto(x,1/y) for y≠0y\ne0; preserve the original domain exclusions.

The full source table continues on PDF 23. Signs stay the same, and increasing/decreasing behaviour reverses on each interval where f is continuous and non-zero. Non-zero local maxima/minima become minima/maxima; intersections of the original and reciprocal occur at f(x)=±1f(x)=\pm1. A zero is excluded and gives a vertical asymptote when the nearby values tend to zero. The source calls zero limits at original vertical asymptotes “x-intercepts”; this needs qualification: excluded inputs remain holes, not actual intercepts, unless an extension is explicitly defined. The source sidebar contrasts reciprocal shape changes with translations/scalings; non-zero affine scaling preserves the number of turning points, but not necessarily metric shape or the original position of a symmetry axis.

How did this part go?